In a crying way, or in a way that is close to crying. You use it to describe HOW someone did something — how they spoke, how they said goodbye, how they thanked you. "She tearfully thanked the doctors" means she was crying, or her voice was shaking and her eyes were wet, while she said thank you. The tears are not the point of the sentence; the thanking is. That is why English puts it as an adverb — it paints the face and the voice of the person doing the action.

Most of the time it means real, honest crying, from sadness, relief, or joy. Note that it is not only for sad news: people tearfully hug at airports and tearfully accept prizes. What it really tells you is that the feeling was too big to keep inside.

It feels a little written — you meet it in news reports, novels, and speeches far more than in daily conversation. A friend telling you a story would more likely say "she was crying when she told me". Because of that news-report flavour, it can sometimes carry a faint hint that the writer is describing a scene from outside, watching the person — so occasionally it is used a bit coldly, or even mockingly, about someone whose tears you do not fully believe ("he tearfully denied everything").